In [6], Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm’s bounds give generators of the Hecke algebra as a module. We also have well-known recursion formulae for the operators with prime. It is the purpose of this paper to prove analogous results in the half-integral weight setting. We also give an explicit formula for how operators commute with the Shimura correspondence.
Dans [6], Shimura a introduit la notion de formes modulaires de poids demi-entier et leurs algèbres de Hecke ; il a aussi établi leur lien avec les formes modulaires de poids entier via la correspondance de Shimura. Pour les formes modulaires de poids entier, les bornes de Sturm permettent de déterminer des générateurs de l’algèbre de Hecke comme module. L’on dispose également de formules de récurrence bien connues pour les opérateurs en les premiers. Le but de cet article est d’établir des résultats analogues dans le cas de poids demi-entier. Nous donnons également une formule explicite sur la commutativité des opérateurs avec la correspondance de Shimura.
@article{JTNB_2014__26_1_233_0, author = {Purkait, Soma}, title = {Hecke operators in half-integral weight}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {233--251}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {26}, number = {1}, year = {2014}, doi = {10.5802/jtnb.865}, zbl = {06304187}, mrnumber = {3232773}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jtnb.865/} }
TY - JOUR AU - Purkait, Soma TI - Hecke operators in half-integral weight JO - Journal de théorie des nombres de Bordeaux PY - 2014 SP - 233 EP - 251 VL - 26 IS - 1 PB - Société Arithmétique de Bordeaux UR - http://www.numdam.org/articles/10.5802/jtnb.865/ DO - 10.5802/jtnb.865 LA - en ID - JTNB_2014__26_1_233_0 ER -
Purkait, Soma. Hecke operators in half-integral weight. Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 1, pp. 233-251. doi : 10.5802/jtnb.865. http://www.numdam.org/articles/10.5802/jtnb.865/
[1] F. Diamond and J. Shurman, A First Course in Modular Forms, GTM 228, Springer-Verlag, 2005. | MR | Zbl
[2] W. Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982), 32–72. | MR | Zbl
[3] T. Miyake, Modular Forms, Springer-Verlag, 1989. | Zbl
[4] S. Niwa, Modular forms of half integral weight and the integral of certain theta-functions, Nagoya Mathematical Journal 56 (1975), 147–161. | MR | Zbl
[5] K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and -Series, CBMS 102, American Mathematical Society, 2004. | MR | Zbl
[6] G. Shimura, On Modular Forms of Half Integral Weight, Annals of Mathematics, Second Series, Vol. 97, 3 (1973), pp. 440–481. | MR | Zbl
[7] W. Stein, Modular forms, a computational approach, Graduate Studies in Mathematics 79, American Mathematical Society, 2007. | MR | Zbl
[8] J. Sturm, On the Congruence of Modular Forms. Number theory (New York, 1984-1985), Lecture Notes in Math. 1240, Springer, Berlin, (1987), 275–280. | MR | Zbl
Cited by Sources: